Absolute Dilation of Fourier Multipliers

Let M be a von Neumann algebra equipped with a normal semifinite faithful (nsf) trace. We say that an operator T: M → M is absolutely dilatable if there exist another von Neumann algebra M with an nsf trace, a unital normal trace preserving -homomorphism J: M → M, and a trace preserving -automorphism U: M → M such that Tᵏ = EJ Uᵏ J \ for all\ k ≥ 0, where EJ: M → M is the conditional expectation associated with J. For a discrete amenable group G and a function u:G→ C inducing a unital completely positive Fourier multiplier Mᵤ: VN(G) → VN(G), we establish the following transference theorem: the operator Mᵤ admits an absolute dilation if and only if its associated Herz–Schur multiplier does. From this result, we deduce a characterization of Fourier multipliers with an absolute dilation in this setting. Building on the transference result, we construct the first known example of a unital completely positive Fourier multiplier that does not admit an absolute dilation. This example arises in the symmetric group S₃, the smallest group where such a phenomenon occurs. Moreover, we show that for every abelian group G, every Fourier multiplier always admits an absolute dilation.